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\(L^{2}\)-rigidity in von Neumann algebras. (English) Zbl 1170.46053

Let \(N\) be a finite von Neumann algebra with trace \(\tau\). If \(M\) is a finite von Neumann algebra with trace \(\tau'\) such that \(N\subset M\) and \(\tau'|_M=\tau\), and \(\delta\) is a densely defined real closable derivation on \(M\) into \((L^2(M,\tau')\otimes L^2(M,\tau'))^\infty\), then the associated deformation \(\{\eta_\alpha\}_\alpha\) coming from resolvent maps is called an \(L^2\)-deformation for \(N\).
If \(B\subset N\) is a von Neumann subalgebra, then the inclusion of \(B\) into \(N\) is called \(L^2\)-rigid if any \(L^2\)-deformation for \(N\) converges uniformly on the unit ball of \(B\). \(N\) is \(L^2\)-rigid if the inclusion \(N\subset N\) is \(L^2\)-rigid. This property generalizes property (T), which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group.
It is proved that, if \(N\) is a II\({}_1\) factor which is not prime or has property \(\Gamma\) of Murray and von Neumann, then \(N\) is \(L^2\)-rigid.
As a consequence, it is shown that, if \(M\) is a free product of diffuse von Neumann algebras, or if \(M=LG\), where \(G\) is a finitely generated group with \(\beta_1^{(2)}(\Gamma)>0\), then any non-amenable regular subfactor of \(M\) is prime and doesn’t have properties \(\Gamma\) or (T).

MSC:

46L10 General theory of von Neumann algebras
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